18 problems

Let $V_{\rho}$ be the invariant set and let $W_{\rho}$ be the local invariant manifold appearing in the construction, with coordinates $(z,v,u,y_1,y_{2:q})$. The $v$-axes are the d…

Local-manifold conjecture. The assumption $\operatorname{D}G(0)=\operatorname{D}^2G(0)=0$ can be removed while still obtaining local rough invariant manifolds.

Consider the equation … near the equilibrium. Conjecture. There are local center-stable and center-unstable submanifolds $M_{\mathtt{s}\mathtt{c}}$ and $M_{\mathtt{u}\mathtt{c}}$ f…

The paper develops invariant-manifold theorems for random parabolic evolution equations with almost sectorial operators, including local stable and center manifolds and conditions…

Let a fast-reaction system be represented in the Fourier domain, and suppose that formal hyperbolicity fails on a set of positive measure in that domain. Invariant-manifold conject…

Let $W^{u,s}_E$ and $W^{u,s}_C$ denote the invariant manifolds associated with the configurations $\mathcal{T}_E$ and $\mathcal{T}_C$ in the $(n+2)$-body problem, and consider thei…

The compactified system is obtained from a nonautonomous system with autonomous future and past limit systems, and the equilibria of those limit systems correspond to equilibria of…

Consider a mode-locked periodic orbit in a three-dimensional map whose stable and saddle periodic points have undergone flip bifurcations. Let their third eigenvalues be…

Let $A^+$ and $A^-$ be general normally hyperbolic compact invariant sets of the respective limit systems. Invariant manifold extension conjecture. The statements for equilibria in…

Let $P_n(x,y)$ and $Q_n(x,y)$ be polynomials of degree $n$, defining the planar polynomial system … An invariant line is a straight line preserved by the flow of this system. Ye's…

Existence conjecture. For every given suitable system, there should exist a coordinate transformation establishing a nearby system that possesses an exact nonlinear normal mode.

In the Sitnikov problem, let $e\in(0,1)$ denote the eccentricity, and in the restricted planar three-body problem (RPTBP), let $\mu\in(0,1/2]$ denote the mass parameter. An invaria…

Let $\Omega_\rho$ be the configuration space of ordered transition points and let $\mathcal{M}^{\textrm{\tiny FH}}$ be the manifold of piecewise-defined profiles $U^{\bm h}$ constr…

The Lomelí map is studied through validated parameterizations of its local stable and unstable manifolds near the fixed points $p_1$ and $p_2$. In the bubble region, consider the g…

Generic transport conjecture. Transverse intersections of stable and unstable manifolds of whiskered tori should be a generic mechanism for transport in phase space and global inst…

Let $f\colon M\to M$ be a $C^1$-diffeomorphism, and let $Orb(p)$ be an orbit with well-defined non-vanishing Lyapunov exponents. In the two-dimensional case, assume that … Here…

Downward influence transmission conjecture. Strong enough $(j+1)$ influences on a limit cycle of level $j$ can transmit along the unstable manifold of this cycle toward the next lo…

Equal-order cycle connection conjecture. The unstable manifold of the second cycle of $S_j$ is generically able to connect to the stable manifold of the second cycle of $S_{j-1}$,…